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Mirrors > Home > MPE Home > Th. List > sspsstri | Structured version Visualization version GIF version |
Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.) |
Ref | Expression |
---|---|
sspsstri | ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | or32 925 | . 2 ⊢ (((𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴) ∨ 𝐴 = 𝐵) ↔ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ⊊ 𝐴)) | |
2 | sspss 4064 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) | |
3 | sspss 4064 | . . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴)) | |
4 | eqcom 2744 | . . . . . 6 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | 4 | orbi2i 912 | . . . . 5 ⊢ ((𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴) ↔ (𝐵 ⊊ 𝐴 ∨ 𝐴 = 𝐵)) |
6 | 3, 5 | bitri 275 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊊ 𝐴 ∨ 𝐴 = 𝐵)) |
7 | 2, 6 | orbi12i 914 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) ∨ (𝐵 ⊊ 𝐴 ∨ 𝐴 = 𝐵))) |
8 | orordir 929 | . . 3 ⊢ (((𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴) ∨ 𝐴 = 𝐵) ↔ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) ∨ (𝐵 ⊊ 𝐴 ∨ 𝐴 = 𝐵))) | |
9 | 7, 8 | bitr4i 278 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ ((𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴) ∨ 𝐴 = 𝐵)) |
10 | df-3or 1089 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴) ↔ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ⊊ 𝐴)) | |
11 | 1, 9, 10 | 3bitr4i 303 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 846 ∨ w3o 1087 = wceq 1542 ⊆ wss 3915 ⊊ wpss 3916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-v 3450 df-in 3922 df-ss 3932 df-pss 3934 |
This theorem is referenced by: ordtri3or 6354 sorpss 7670 sorpssi 7671 funpsstri 34379 |
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